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Q2(ii):
Find :
(ii) $32^{\frac{2}{5}}$
Solution :
Given Expression & Initial Setup
We are required to evaluate the exponential expression:
$32^{\frac{2}{5}}$
This expression features a base of $32$ raised to a rational (fractional) exponent $\frac{2}{5}$. To simplify this, we must decompose the base into its fundamental prime factors.
Step 1: Prime Factorization of the Base
We begin by finding the prime factorization of the integer $32$. By continuously dividing by the smallest prime number ($2$), we can express $32$ as a power of $2$.
- $32 \div 2 = 16$
- $16 \div 2 = 8$
- $8 \div 2 = 4$
- $4 \div 2 = 2$
- $2 \div 2 = 1$
Thus, the prime factorization is:
$32 = 2 \times 2 \times 2 \times 2 \times 2 = 2^5$
Step 2: Substitution and Application of Exponent Laws
Substitute the prime factorized form ($2^5$) back into the original expression in place of $32$:
$ (2^5)^{\frac{2}{5}} $
To simplify this, we apply the Power of a Power Property of exponents [Per the fundamental law of exponents which states that $(a^m)^n = a^{m \cdot n}$]. This allows us to multiply the inner exponent by the outer rational exponent:
$ 2^{5 \times \frac{2}{5}} $
Step 3: Simplification of the Exponent
Now, perform the multiplication in the exponent. The integer $5$ in the numerator cancels out perfectly with the denominator $5$ of the fraction:
$ 5 \times \frac{2}{5} = \frac{5 \times 2}{5} = \frac{10}{5} = 2 $
Substituting the simplified exponent back onto the base of $2$ yields:
$ 2^2 $
Step 4: Final Evaluation
Evaluate the simplified exponential expression by squaring the base:
$ 2^2 = 2 \times 2 = 4 $
Final Solution: 4
More Questions from Class 9 Mathematics Number Systems EXERCISE 1.5
- Q1(i): Find : (i) $64^{\frac{1}{2}}$
- Q1(ii): Find : (ii) $32^{\frac{1}{5}}$
- Q1(iii): Find : (iii) $125^{\frac{1}{3}}$
- Q2(i): Find : (i) $9^{\frac{3}{2}}$
- Q2(iii): Find : (iii) $16^{\frac{3}{4}}$
- Q2(iv): Find : (iv) $125^{-\frac{1}{3}}$
- Q3(i): Simplify : (i) $2^{\frac{2}{3}} \cdot 2^{\frac{1}{5}}$
- Q3(ii): Simplify : (ii) $(\frac{1}{3^3})^7$
- Q3(iii): Simplify : (iii) $\frac{11^{\frac{1}{2}}}{11^{\frac{1}{4}}}$
- Q3(iv): Simplify : (iv) $7^{\frac{1}{2}} \cdot 8^{\frac{1}{2}}$
CBSE Solutions for Class 9 Mathematics Number Systems
Chapters in CBSE - Class 9 Mathematics
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